Phys 332 Electricity & Magnetism Day 3. Note: I should have recommended reading section 1.5 (delta function) as well. rˆ rˆ

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1 Phs 33 lecticit & Magnetism Da 3 Mn. 9/9 Wed. 9/ Thus 9/ Fi. 9/3 (C.-.5,.8). &.5;..-.. Gauss & Div, T Numeical Quadatue (C.-.5,.8)..3 Using Gauss (C.-.5,.8) Using Gauss HW quipment Bing in ppt s Gauss s La Tutial Nte: I shuld have ecmmended eading sectin.5 (delta functin) as ell. Last Time Last time e ked n cmputing the electic field at sme bsevatin lcatin due t a cntinuus distibutin f chage. In geneal, that meant summing, i.e., integating, the cntibutins f all the diffeential msels f chage: Q dq ˆ Depending n the gemet f the chage distibutin, this sum ill be paameteied in diffeent as s that athe than summing ve chage, u e summing ve lcatins that the chage ccupies. Fail geneall, dq ˆ ˆ d V V This Time Unftunatel, that integal can be a challenge t pefm. F sme gemeties, unde sme appimatins, thee s a a that s ften simple using Gauss s La. Back in Phs 3, e deived S da Q enc Which e call Gauss s La. The agument ent smething like this: Flu Fist e gt familia ith the ntin f Flu, geneall a measue f fl thugh an aea / int ut f a vlume. ample: Rain Flu thugh single, pen aea. Sa it s aining ut and u have left u ind pen. Then, a easnable questin uld be, at hat ate is ain cming in u ind. T make it cncete, let s sa that e measue

2 Phs 33 lecticit & Magnetism Da 3 amunt f ain in tems f the mass f. What s the ate at hich cmes thugh the ind? v dl= v h h ind ind ind dm m Vl vacs v A dvl dl h dl h v hcs Flu thugh hle clsed aea. N, hat if e ee inteested in nt just the flu thugh a paticula ind, but int the hle m. That s simpl the sum f flues in thugh all inds, ut thugh the d, dn thugh the fl (b, that s gnna be a mess t clean up!). m ind ind fl d v i i A i Aea diectin cnventin in. In this case, it s cnvenient t have all the aea vects pint int the m, s that u get psitive cntibutin if the velcit pints in and negative cntibutin if it pints ut (as ith the d and fl.) Integal Fm. Me geneall, this discete sum can be itten as a cntinuus integal. v. int. m F the sake f futue aguments, ften ne talks abut the flu ut thugh a clsed aea athe than in thugh it. Of cuse, that s just the ppsite f in thugh the aea. v. ut. m in ut Genealiing: An vect field. While u phsics idea f flu cnnects best ith the cmmn usage hen e e talking abut smething (in this case, ) being tanspted, the same mathematical and cnceptual tl can be applied t an thing that s epesented b a vect field, such as the electic field.

3 Phs 33 lecticit & Magnetism Da 3 lectic Flu N let s appl the same idea t an lectic field. Thugh a patch f aea, A In ut f a clsed suface As Giffith s pints ut, the analg t steam lines is electic field lines. Mtivatin What s it gd f? That s a sell definitin and all, but u ma be ndeing h e bthe hat pactical use is such a definitin. It tuns ut that thee s a ve simple elatin beteen electic flu and chage distibutin. That means that if u kn ne, it s eas t find the the, and in sme cases it s easie t slve f thugh a flu agument than simpl summing ve the suces as e ve dne in the past. If that seems a th gal, then let s figue ut hat that but hat s imptant is that, egadless f h mess the integal is, it has ve simple slutin. The bk agued it ut piece b piece, s n let s put all the pieces tgethe in a cheent st. Relate Flu and Chage This is a ve geneal tl, s I ll give u sme vague visuals t help think abut it, but dn t take them t liteall. Sa e have sme chage distibutin like ant t kn the Flu f the electic field ut a suface like this q q 5 q q q 3 Fist, Suppepsitin: ecall that the field f a chage distibutin is simpl the sum f the fields due t each pint chage that makes up the distibutin. That is, n matte h u have N chages distibuted, the field at a given lcatin n the suface (net) is simpl 3... N q 6 q q q q 3 q 5

4 Phs 33 lecticit & Magnetism Da 3 That means that N... N Lking just at Chage (hich is inside the aea). S, e can divide and cnque find the flu due t ne pint chage, and then put it back tgethe. Of cuse, the field at u q bsevatin lcatin due t a pint chage is simpl ˆ. S, q q q ˆ ˆ ˆ N, let s cnside that -pduct. It simpl calls f the pjectin f the patch f aea pependicula t -hat. S, even if e e dealing ith a suface like 3... N nˆ q e still nl have t abut the pependicula pjectin f the patch f aea. Let s paameteie the patch in tems f vaiables e can hpe t integate. Given the spheical smmet f the pint chage s electic field, a gd chice f cdinate sstems is spheical. In that sstem, a diffeential aea is d sin d d sin Plugging that int u flu equatin, e have Flu thugh patch. d. N, acss just this little patch f suface aea, the flu is then just q q sin sin Depends nl n Angles. Remakabl, it nl depends n the slid angle the aea subtends.

5 Phs 33 lecticit & Magnetism Da 3 If the slid angel f the aea subtends is a bit feign t u, think f it this a. Sa u e a chage, adiating in all diectins, and this m, it s alls, ceiling, and fl define the suface aea. H much f u adiatin passes thugh a given suface depends n h much f u vie it takes up. Lking at a tile abve u, it measues abut ( /8 adians) b abut ( /8 adians) s its slid angle is ughl the pduct, /3 ad, that s hat detemines h much f u adiatin fls thugh it. Flu thugh hle suface. Ging ahead and integating ve the hle suface aea then gives q q q q d sin d d sin d And thee s a ve simple esult! Q: What abut chage n the utside? Chage utside the aea. N, befe e g back t the ttal flu f a big chage distibutin, e need t cnside the the case hee e ve cnsideed a pint inside the suface, hat abut ne utside the suface? Well, ntice that the -pduct f and n as psitive, and that, as lng as the chage is inside the suface, it and n ill alas bth pint utad, s the ll alas be psitive. nˆ q If the chage is utside, n the nea suface n and ill pint in ppsite diectins, s the pduct ill be negative hile n the fa side the ll pint in the same diectin, s the ll be psitive. All that mattes in the math is the angle subtended; pehaps u ll bu that, as u seep thugh angles, f eve patch thugh hich the field fls ut thee s a canceling patch thugh hich the field fls in. S summing ve the hle suface, the flu cmes t e. (This is a ve sketch agument, the bk is a little me thugh.) The analgus situatin uld be if u had a she head spaing ut in all diectins (instead f a chage) and a lse-mesh bag (f the suface), then hateve fls in fm the left, fls ut thugh the ight, making f n net flu int ut f the bag. 5

6 Phs 33 lecticit & Magnetism Da 3 S an chage enclsed b the suface cntibutes t the flu t the tune f ecluded chages cntibute nthing. q enclsed hile all Thus e have q Q enclsed q 3 q... egadless f the shape f the suface. This is the integal fm f Gauss s La. N We als deived the diffeential fm f this. See PePint () Gauss s la Cnside a small b ith edges alng the cdinate aes. ( ),, ( ) Calculate the electic flu pe vlume in the it that the vlume ges t e, hich is the divegence f : Divegence 6 Mtivatin. Retun t Rain Recall that the geneal idea f a flu is a fl ate: the chage flu dn a ie, dq/, is the cuent. Similal, in the eample f ain that e used t mtivate the definitin f flu, the ate at hich entes a m thugh sme pen dm inds uld be a flu. v Nmaliing pe Vlume. N, if I tld u that kg f ained in pe minute, u d be pett ied until I tld u that the m as the Supedme- that vlume s huge. kg / minute leak isn t s bad as if e ee talking abut, sa, this m. This eample illustates that flu alne desn t tell the hle st. Smetimes u e me inteested in flu pe vlume. On a pe vlume basis, the same flu int the Supedme is nthing cmpaed t that int

7 Phs 33 lecticit & Magnetism Da 3 Math. div this m. Flu ut pe vlume is Divegence. (Cnvesel, I suppse e d call Flu in pe vlume Cnvegence = Divegence) div Vl N f a little math. V V ˆ n V dm Vl v Vl The the side f Gauss s la ve the vlume in the it that the vlume ges t e is: V q inside V, hee is the chage densit. The diffeential fm f Gauss s la is: div Nte that this is a scala equatin. In the secnd fm, the del peat is i ˆ ˆ j ˆ k. amples/ecises: Pblem (fm anse f.6) Suppse the electic field (in clindical cdinates) is Csˆ s s a Ca s s ˆ s a What is the chage densit in each egin? The chage densit is. Since the electic field nl has an s (adial) cmpnent, the divegence (fm the fnt cve) is 7

8 Phs 33 lecticit & Magnetism Da 3 s Cs C s a s s s Ca s s a s s s Cs Cmputatinal Tutial # Numeical Integatin (have students g thugh it) Pevie The fist attempt at HW # is due tm at 3 pm. Yu must make a fist attempt in de t get an cedit f a pblem! F Mnda, u ll ead abut appling Gauss s La. 8